Dielectric relaxation and charge carrier mechanism in nanocrystalline Ce–Dy ionic conductors

Abstract

Ion dynamics in pure and Dy containing nanoceria has been investigated in the light of different defect associates and their mutual interactions. The samples were prepared through citrate–nitrate auto-ignition method and their single-phase cubic fluorite structure was confirmed from X-ray diffraction and high resolution transmission electron microscopy analysis. The complex impedance spectra showed both grain and grain boundary contribution to total conductivity. The concentration dependent conductivity variation has been discussed with the help of oxygen vacancy concentration and their interactions with the defect associates. The frequency dependence of dielectric permittivity and electric modulus has been analyzed using Havriliak–Negami formalism. The relaxation mechanism is found to be dependent on the formation of different dimers and trimers. Modulus analysis has established the charge re-orientation relaxation of the defects associates. The time–temperature superposition principle has been established by scaling of different spectra.

---

1. Introduction

Nanostructured doped ceria electrolytes have gained great interest with a potential application in intermediate temperature solid oxide fuel cells (IT-SOFCs) due to their high performance at intermediate temperatures.¹ In doped ceria, the oxygen ion conductivity mainly depends upon the nature and concentration of dopant ions, oxygen vacancies and local defect structure. When a rare earth is doped in a ceria lattice, an ionic conductivity originates by migration of oxygen ions through oxygen vacancies because of the substitution of tetravalent Ce ions by trivalent rare earth ions.² Therefore, to achieve a higher ionic conductivity for oxygen ions, an availability of oxygen vacancies inside the electrolytes is significant.^3,4 The major problem of these solid electrolytes is the increasing electronic conductivity at higher temperatures and lower oxygen partial pressure because of the reduction of Ce⁴⁺ to Ce³⁺ and also segregation of impurities at the grain boundaries.^5,6 For a smaller grain size, there is a probability of spread of the impurities over a large interfacial area and therefore the effect of impurities can be reduced. So, it is assumed that, the grain boundary may provide a faster ionic diffusion pathway for defects resulting in enhanced ionic conductivity and nanoscale doped ceria can be used at lower temperatures to avoid the reduction of Ce⁴⁺ to Ce³⁺.⁷ The oxygen vacancies are not necessarily free but may be bound to the doped cations forming defect associate such as due to columbic interaction.⁸ Among rare earth dopants, ceria doped with Sm³⁺ and Gd³⁺, shows higher conductivity because of their optimum ionic radius and thereby a smaller association enthalpy.^9,10 According to T. Mori et al.,¹¹ an appropriate processing route can improve conductivity of Dy³⁺ doped ceria comparable with Sm³⁺ and Gd³⁺ doped ceria. The dielectric properties of rare earth doped ceria materials depends on these defect associates. According to Sarkar and Nicholson,^12,13 in lightly rare-earth doped ceria, there are relaxation processes which are due to the orientation relaxation of the charged associated defects. Although the oxide ion conduction may be affected by the defect associate, the relationship between the oxide ion conductivity and the associate pairs has not been investigated in details. Most of the studies concerning the relationship between oxide ion conduction and dielectric properties have been carried out in relatively lower doping concentration. The dielectric relaxation of nanocrystalline ionic conductors was reported by many authors^14–16 but the dielectric relaxation of rare earth doped ceria is limited. Previously, some authors discussed temperature dependence of dielectric relaxation for rare earth doped ceria.^17,18 The effect of dopant concentration on the conductivity and relaxation due to formation of oxygen vacancy and defect associates was not investigated previously. H. Yamamura et al.¹⁹ investigated the dielectric relaxations for the Ce–Nd system. They only pointed out the presence of two kinds of Debye-type relaxation due to charged dimmers and neutral trimers. Recently, S. A. Acharya²⁰ reported that the dielectric relaxation of the grain and the grain boundary were found to be dependent on their size and associated with the charge conduction mechanism for the Ce–Dy system. In both of the above studies the role of defect associates and oxygen vacancies on conduction and relaxation mechanism has not been considered.

Therefore, in our present work, a wide composition range of Ce_1−xDy_xO_2−δ (0 ≤ x ≤ 0.50) nanoparticles are prepared by citrate auto-ignition method and we discuss the role of defect associates and oxygen vacancies on the conduction and relaxation mechanism in Ce–Dy ionic conductors. The dependence of conductivity on oxygen vacancies and defect associates are discussed and co-related. The relaxation processes have been analyzed using the Havriliak–Negami equation. The time–temperature superposition principle has been verified for this system.

2. Materials and methods

Dy doped ceria nanoparticles, Ce_1−xDy_xO_2−δ (0 ≤ x ≤ 0.50) were prepared using low temperature citrate–nitrate auto-ignition method. Ce(NO₃)₃·6H₂O (99.9%) (MERCK) and Dy₂O₃ (Sigma-Aldrich) (99.9%) were used as starting materials. The details of sample preparation process had been discussed in our earlier work.²¹ The as prepared powder samples were annealed at 400 °C for 2 h and then sintered at 600 °C for 6 h. During annealing and sintering, the rise in temperature was at the rate 5 °C min⁻¹ and the cooling was normal cooling. X-Ray diffraction profiles of the samples were recorded with a powder X-ray diffractometer (BRUKER, Model D8 Advance-AXS) using CuK_α radiation [λ = 1.5406 Å] from 2θ = 20° to 90° with step size 0.05°. The microstructural analysis of the samples at high magnification were performed by placing the particles in a formvar–carbon coated 300 mesh copper grid with the help of transmission electron microscope (JEOL, Model JEM-2010) operated at 200 kV. For electrical measurements, cylindrical pellets were prepared from sintered powder by uniaxial pressing in a 10 mm diameter stainless steel die. The pellets were sintered again at 600 °C for 6 hours for densification. The relative densities of the sintered pellets were measured using Archimedes's principle and found that values were in between 93% and 97%. The pellets were covered on both sides with conductive graphite paste (Alfa-Aesar) to make the electrodes. The electrical measurements were performed in a tube furnace using two probe methods in air. An LCR meter (HIOKI, MODEL 3532-50) interfaced with PC, was used to collect the electrical data in the frequency range 42 Hz to 5 MHz and in the temperature range 250–550 °C.

3. Results and discussion

3.1. X-Ray diffraction analysis

The crystal structure and phase purity of Dy³⁺ doped ceria have been confirmed by XRD analysis using the Rietveld structure refinement process with the help of MAUD software (version 2.33). The Rietveld refined XRD patterns of the nanopowders are shown in Fig. 1. The analysis confirms the single phase cubic fluorite structure with space group Fm3̄m for each sample. The crystallites sizes of the system vary in the range 13.49 nm to 21.65 nm. A detailed structural analysis of the same system has been given in our earlier study.²² A lattice expansion of the system had been observed with concentration of Dy³⁺ ions because of substitution of smaller Ce⁴⁺ (0.97 Å) by larger Dy³⁺ (1.027 Å) ions. The lattice expansion at higher doping concentration is relatively smaller than the lower doping range. This can be attributed to the effect of columbic interaction between dopant cations and oxygen vacancies. The Rietveld analysis also confirmed the increase of oxygen vacancies with the doping concentration.²²

Fig. 1 The Rietveld refined XRD patterns of the prepared samples. Experimental data are shown by black hollow circle, simulated patterns are shown by solid red lines and lower blue line represents the corresponding difference for each of the pattern. Markers show peak positions.

3.2. Transmission electron microscopy (TEM) study

To get insight into the microstructure of the samples, investigation using HR-TEM was performed. Fig. 2 shows the HR-TEM micrograph of the sample Ce_0.8Dy_0.2O_2−δ. It shows good crystallinity with clear resolution of fringes of lattice planes (some of them are marked by white short parallel lines). The inter-planar distances of different lattice fringes have been measured which correspond to (111), (200), (220) and (311) planes of the sample. The corresponding SAD pattern of the sample is shown in the inset of Fig. 2. The hkl values of different planes have been evaluated from the rings of the SAD pattern which again confirms the cubic fluorite phase of the sample. It can be seen in Fig. 2 that, in some zones (indicated by (a) to (d)), the lattice slightly deviates from the exact fluorite structure matrix which can be evidenced by the lattice dislocation. The zones (a) and (c) show the edge dislocation at the grain boundary region. In zones (b) and (d), a screw dislocation has been generated. A closer view of this micrograph also shows some other zones consisting of the above mentioned lattice distortions. These types of lattice dislocation are also present in the other samples in our study. Fig. 3(a) shows the STEM image and Fig. 3(c–e) shows the Ce L_a, Dy L_a and O K_a elemental mapping of the sample Ce_0.8Dy_0.2O_2−δ. The image in the pink square box represents the CeO₂ nanoparticles which provide a reference to locate the area of the elemental mapping. The mapping has been obtained as a pixel-by-pixel mapping of the integrated intensity. This elemental mapping confirms the uniform presence of Ce, Dy and O throughout the nanostructure regions. It also indicates the incorporation of Dy into the ceria lattice and absence of Dy as a free species within the nanostructure. The corresponding EDAX spectra in Fig. 3(b) also confirms the above inference.

Fig. 2 The HR-TEM micrograph of the sample Ce_0.8Dy_0.2O_2−δ. The corresponding SAED pattern of the sample is shown in the inset.

Fig. 3 (a) The STEM image, (b) energy dispersive spectroscopy and (c–e) the elemental mapping for the composition Ce_0.8Dy_0.2O_2−δ.

3.3. Impedance spectroscopy study

Fig. 4 shows the complex impedance plots (Nyquist plot) for the composition Ce_0.8Dy_0.2O_2−δ at different temperatures. This plot shows two successive depressed semicircular arcs. The high and low frequency semicircular arcs correspond to grain and grain boundary contribution respectively.²³ Thus, the total conductivity has both the grain and grain boundary contributions. The typical equivalent circuit for fitting of such impedance plots is given in the inset of Fig. 4. It consists of a series combination of two resistors R₁ and R₂ in parallel with constant phase elements Q₁ and Q₂. Subscripts 1 and 2 represent the grain and grain boundary respectively. Using this equivalent circuit in EC-Lab software we have fitted the complex impedance spectra and the corresponding fitting equation isQ represents the deviation of capacitance from ideal behavior. The true capacitance (C) can be calculated from the expression:where the exponent a lies between 0 and 1. The value of the exponent ‘a’ is very significant since 0 < a < 1 indicates the presence of non-Debye type relaxation in the sample. The values of these fitting parameters (R, Q and a), obtained after fitting, are listed in Table 1. Here, the value of Q increases with temperature while the values of both R₁ and R₂ decrease. It has also been found that, the grain conductivity is ∼2–3 orders of magnitude higher than the specific grain boundary conductivity. The reciprocal temperature dependence of grain and specific grain boundary conductivities are shown in Fig. 5 which obeys the Arrhenius equation given by,where the symbols have their usual meaning. Fig. 5 also shows that, the conductivity increases with temperature indicating that the oxygen ion conduction in grain and grain boundary is a thermally activated process. This increase in conductivity with temperature is attributed to the increase of oxygen vacancies that are free to migrate.²⁴ The variation of activation energy and conductivity at a particular temperature, for both the grain and grain boundary with the doping concentration of Dy³⁺ ions (x) are shown in Fig. 6. The grain conductivity of all the Dy doped ceria is higher than the pure ceria. The specific grain boundary conductivity of pure ceria is higher than the Dy doped ceria upto x = 0.40 and at high doping concentration its value is lower than pure ceria. The activation energy of pure ceria is higher than the Dy doped ceria upto x = 0.25 for grain conductivity and upto x = 0.40 for specific grain boundary conductivity. However, it can be observed that, the conductivity increases with x and reaches a maximum value for x = 0.20. After that, conductivity decreases with x while the activation energy shows an opposite behavior. The peculiarity can be explained as follows.

Fig. 4 The complex impedance plot for the composition Ce_0.8Dy_0.2O_2−δ at different temperatures.

Table 1 The values of resistances (R), constant phase elements (Q) and exponent (a) as obtained after fitting for the composition Ce_0.8Dy_0.2O_2−δ. The errors are indicated in the parenthesis

Temperature	Q1 (F s^a1−1) × 10^−11	a1 (±0.02)	R1 (Ω)	Q2 (F s^a2−1) × 10^−8	a2 (±0.02)	R2 (Ω)
550 °C	89.42	0.91	385	20.68	0.92	213
525 °C	80.47	0.92	580	26.94	0.95	309
500 °C	57.11	0.94	924	33.89	0.97	462
475 °C	42.23	0.96	1558	28.47	0.95	778
450 °C	40.14	0.97	1725	29.39	0.98	1365

---

Fig. 5 The reciprocal temperature dependence of (a) grain and (b) specific grain boundary conductivity.

Fig. 6 The variation of activation energy and conductivity at 550 °C for (a) grain and (b) specific grain boundary with the doping concentration of Dy³⁺ ions.

When Dy³⁺ ions are doped into ceria, then for charge neutrality, one oxygen vacancy is formed for every two Dy³⁺ ions, which may be represented by the Kröger–Vink notation:

here Dy^′_Ce indicates one Ce⁴⁺ site occupied by Dy³⁺ ion and is the oxygen vacancy. In our earlier study,²² it had been shown that, oxygen vacancy increases with doping concentration of Dy³⁺ ions. Thus, with the increase of Dy³⁺ ion concentration the conductivity increases and reaches a maximum value with minimum activation energy at x = 0.20. After that, due to the increasing interactions between Dy³⁺ ions and oxygen vacancies and formation of local defect structures (which lowers the mobile oxygen vacancies), the conductivity decreases.²⁵ For lower concentration of Dy³⁺ ions, the association enthalpy is mainly due to the formation of dimers . Similarly, for higher concentration of Dy³⁺ ions, there is a high probability of forming trimers as the concentration of Dy³⁺ ions grows twice that of the oxygen vacancies that can be represented as:

This formation of trimers and local defect structures increases with Dy³⁺ ion concentration. Usually, high thermal energy is required for the oxygen vacancies to overcome the dopant interaction barrier²⁶ which accounts for higher activation energy and lower conductivity.

3.4. Dielectric spectroscopy study

The frequency variation of the real part of the complex dielectric function (ε′(ω)) at different temperatures for the sample Ce_0.8Dy_0.2O_2−δ is shown in Fig. 7(a). Fig. 7(b) represents the frequency variation of ε′(ω) for all the compositions at a specific temperature. We have analyzed the frequency dependence of the complex dielectric spectra using the generalized Havriliak–Negami (HN) formalism.²⁷ The generalized dielectric function is given by,where ε_s and ε_∞ are the relaxed and unrelaxed permittivity respectively and their difference ε_s − ε_∞ represents the dielectric relaxation strength with and . α and β are the shape parameters satisfying the condition 0 ≤ α ≤ 1 and 0 ≤ αβ ≤ 1. τ_HN is the characteristic relaxation time. The parameters α and β are related to the limiting behavior of the complex dielectric function at low and high frequencies as:

ε_s − ε′(ω) ∼ ω^α and ε′′(ω) ∼ ω^α for ωτ_HN ≪ 1 (7)

ε′(ω) − ε_∞ ∼ ω^−αβ and ε′′(ω) ∼ ω^−αβ for ωτ_HN ≫ 1 (8)

Fig. 7 (a) The frequency variation of the real part of the complex dielectric function at different temperatures for the sample Ce_0.8Dy_0.2O_2−δ, (b) for all compositions at temperature 525 °C, (c) the variation of ε′′(ω) with frequency for the sample x = 0.20 at different temperatures. The blue straight line represents conduction part and HN dielectric loss peak is shown by red dotted line. (d) The variation of ε′′ with doping concentration.

The real and imaginary parts of ε*(ω) are respectively expressed as,

and

We used a built-in function of eqn (9) in OriginLab software platform. Using this function in OriginLab software the real part of the complex dielectric spectra was fitted. The values of different parameters obtained from the fits are shown in Table 2. This table clearly reveals that the value of ε_∞ and ε_s − ε_∞ is lower and the value of τ_HN is higher for pure ceria than the Dy doped ceria. The Fig. 7(a) and (b) show that, in the low frequency region, ε′(ω) exhibits a high value which is due to the polarization of oxygen ions at the electrode–electrolyte interface and during HN fit we have ignored this electrode polarization. The value of ε′(ω) decreases with increase in frequency and saturates at high frequency region giving rise to ε_∞. The capacitive effect at high-energy sites disappears resulting in the reduction of the contribution of charge carrier to the dielectric constant and decrease of ε′(ω) with frequency.¹⁸ As shown in Fig. 7(a), the real part of dielectric permittivity ε′(ω) exhibits a step (indicated by arrow mark) in the intermediate frequency range. This step is due to the grain boundary relaxation, which leads to the formation of plateau in the dielectric permittivity ε′(ω) in the intermediate frequency range.²⁸ With the rise in temperature, the plateau shifts towards the high frequency region indicating that the grain boundary relaxation is a thermally activated process. Therefore, there exists a change or relaxation of lattice at the grain boundaries, which is associated with the migration of charge carriers there. It also reveals that, the plateau shifts towards the high frequency region with increasing concentration of Dy³⁺ ions and reaches a maximum shift for x = 0.20 and then reverses the direction towards lower frequency region with further increase of Dy concentration. The variation of grain boundary conductivity with doping concentration is similar in nature of the shift of this plateau. Using HN fit, it has been found that, the value of dielectric constant (ε_∞) and dielectric relaxation strength (ε_s − ε_∞) increases with temperature. This increase of ε_∞ is due to the increase in polarization of charge carriers with temperature. Similar behavior is also found for all the other compositions. Fig. 7(c) shows the variation of ε′′(ω) with frequency for the composition x = 0.20 at different temperatures. This spectrum has been fitted using a built-in function of eqn (10) in OriginLab software platform. The term that appears in the imaginary part of ε*(ω) is the conduction part where S is related to the dc conductivity arising from the ionic conduction and p is the frequency exponent. The unity value of p indicates the ideal ohmic behavior.²⁹ We separately fitted the conduction and dielectric part of ε′′(ω) spectra. The blue straight line represents conduction part in the ε′′(ω) spectra. A single Havriliak–Negami dielectric loss peak is shown in Fig. 7(c) by red dotted line. Fig. 7(c) also shows high value of ε′′ at low frequency region which may come from the free oxygen vacancy contributing to dc conduction and the bound which oscillate out of phase with the applied time varying electric field.³⁰ Fig. 7(d) shows the variation of ε′′ with doping concentration at temperature 500 °C and at frequency 7.19 KHz which exhibits a highest value of ε′′ for the composition x = 0.20. The value of ε′′ is lower than it, both at lower and higher doping concentration. The value of ε′′ also shows lowest value for pure ceria and the of ε′′ for x = 0.50 is very close to pure ceria. All the parameters ascribed in HN equation are found to vary with temperature this indicates that the conduction and relaxation mechanism are thermally activated.

Table 2 The values of unrelaxed permittivity (ε_∞), dielectric relaxation strength (ε_s − ε_∞), characteristic relaxation time (τ_HN) and shape parameters (α and β) for all compositions at temperature 525 °C and migration energy (E_m) for all compositions. The errors are indicated in the parenthesis

Composition	ε∞ (± 0.5)	εs − ε∞	τHN (s)	α (± 0.01)	β (± 0.01)	Em (eV) (± 0.01)
CeO2	216	975	1.5 × 10^−2	0.85	0.65	1.20
Ce0.9Dy0.1O2−δ	504.73	7.88 × 10^4	2.00 × 10^−4	0.96	0.97	1.06
Ce0.85Dy0.15O2−δ	427.35	1.23 × 10^4	1.20 × 10^−4	0.92	0.97	1.05
Ce0.8Dy0.2O2−δ	510.95	9.24 × 10^4	6.00 × 10^−5	0.98	0.98	1.02
Ce0.75Dy0.25O2−δ	554.02	9.04 × 10^4	1.00 × 10^−4	0.94	0.98	1.06
Ce0.7Dy0.3O2−δ	399.91	8.84 × 10^4	1.30 × 10^−4	0.98	0.97	1.06
Ce0.6Dy0.4O2−δ	279.51	7.79 × 10^4	1.10 × 10^−3	0.92	0.98	1.10
Ce0.5Dy0.5O2−δ	421.06	4.61 × 10^4	9.15 × 10^−3	0.91	0.97	1.15

---

The frequency dependence of dielectric loss tangent for the composition Ce_0.8Dy_0.2O_2−δ at different temperatures is shown in Fig. 8(a). It shows the existence of a single relaxation peak, which may be attributed to the dipole moment of the defect pair . At higher frequency, the value of tan δ becomes independent of frequency because the dipoles are not able to respond and reorient themselves with the applied frequency. The relaxation peak at resonance frequency f_max shifts towards the higher frequency side and the intensity of the relaxation peak increases with the temperature. The increase of relaxation peak intensity corresponding to defect pair with temperature is due to the increase of oxygen vacancy and defect pair which may be free from defect trimers ³¹ and represented as:

Fig. 8 (a) The frequency dependence of dielectric loss tangent for the sample Ce_0.8Dy_0.2O_2−δ at different temperatures. The inset figure shows temperature dependence of f_max. (b) The frequency dependence of dielectric loss tangent for all the compositions at the temperature 550 °C. The inset figure shows the variation of f_max and tan δ_max with doping concentration.

These oxygen vacancies perform local motions around Dy³⁺ dopant and long-range migration giving rise to higher grain conductivity. The temperature dependence of f_max is shown in the inset of Fig. 8(a), which follows the Arrhenius relation given by

where f₀ is the pre-exponential factor and E_m is the migration energy of oxygen vacancy present on the defect associates or the energy associated with the dipolar motion of . The value of migration energies for all others samples are listed in Table 2. The migration energy for the sample x = 0.2 was found 1.02 eV and this value is slightly higher for the same sample as reported earlier (0.92 eV).⁷ The migration energy for pure ceria is 1.20 eV which is higher than the any Dy doped ceria upto x = 50. According to P. P. Dholabhai et al.³² the activation energies for vacancy migration depends on migration path and activation energy for migration from second nearest neighbor to first nearest neighbor is less than activation energy for migration from first nearest neighbor to second nearest neighbor. Therefore, difference in migration energies may due to the different migration paths. Fig. 8(b) represents the frequency dependence of dielectric loss tangent for all the compositions at a particular temperature. The variation of f_max and tan δ_max (intensity of relaxation peak) at 550 °C with doping concentration of Dy³⁺ ions is shown in the inset of Fig. 8(b). The value of f_max shifts towards higher frequency side with the increase of doping concentration and reaches a maximum value for the composition x = 0.20. Further increase of doping concentration reverses the direction and makes it shift towards lower frequency side. According to H. Yamamura et al.,³³ relaxation peaks at higher frequency side correspond to defect pair such as and the relaxation peaks at lower frequency side are due to neutral trimers such as . Therefore with the increase of Dy³⁺ ion concentration, the number of defect pairs increase, as a result the relaxation peak shifts towards higher frequency side up to x = 0.20. Further increase of Dy³⁺ ion concentration enhances the formation of defect trimers as we discussed earlier, consequently shifting the relaxation peak towards lower frequency side. However, a neutral trimer, giving rise to the dielectric relaxation peak is not linear but a bended associate, because a linear trimer will not show any effective dipole moment and hence there will be no relaxation peak. The shifting of relaxation peak towards lower frequency side at higher doping concentration implies a higher relaxation time and high columbic interaction between oxygen vacancy and associated dopants. The increase in the value of tan δ_max with the Dy doping concentration indicates the increase of oxygen vacancy with doping concentration but these vacancies may not be free.

3.5. Modulus spectroscopy study

Fig. 9(a) shows the complex modulus plots for all the compositions at a particular temperature. The figure shows a single depressed semicircular arc which suggests the presence of electrical relaxation phenomenon in the compositions. It also confirms the single phase formation of all the compositions.³⁴ The intersection of the semicircular arc with the abscissa gives the inverse of the total capacitance. The change in shape of the spectra with doping concentration points to compositional dependence of the relaxation phenomena. Fig. 9(b) and (c) show the variation of real and imaginary parts of complex modulus for all the compositions at a particular temperature respectively. We have fitted the modulus spectra using empirical Havriliak–Negami equation:²⁷where M_∞ is the inverse of high frequency dielectric constant and 0 ≤ μ, ν ≤ 1 are the shape parameters. We divided M*(ω) into the real and imaginary parts like of ε*(ω). The real and imaginary part of modulus spectra have been fitted using a built-in function in OriginLab software platform. The values of different parameters, obtained after fitting, are given in Table 3. As shown in Fig. 9(b), the value of M′(ω) starts from zero in the lower frequency region and shows a continuous dispersion with frequency having a tendency to saturate at a maximum asymptotic value in the higher frequency region. This may be due to a short range mobility of charge carriers and lack of restoring force governing the mobility of charge carriers under the action of an induced electric field.³⁵ The dispersion region shifts towards the higher frequency region with doping concentration up to x = 0.20 and after that, the dispersion region reverses towards the lower frequency region with further increase of doping concentration. All the compositions show a similar variation of M′(ω) with frequency in the entire temperature range. Fig. 9(c) shows that, the value of M′′(ω) approaches to zero at lower frequencies which indicates that, no significant contribution of electrode polarization is present in the modulus spectra. This figure also shows a single relaxation peak for all the compositions which is possibly due to the charge re-orientation relaxation of defects associated with .²³ The oxygen vacancy bound to Dy³⁺ ion, can occupy any one of the eight equivalent sites around Dy³⁺ ion in the cubic fluorite structure of ceria and jumps from one site to the other giving rise to re-orientation of relaxation process.⁷ This jump of bound oxygen vacancies in re-orientation process is similar to that of a free state. Therefore, the activation energy for the re-orientation process in the bound state is analogous as the migration energy of oxygen vacancy in the free state of long-range motion.³⁶ The activation energy for re-orientation process or the activation energy of relaxation has been obtained from the slope of Arrhenius plot of log τ_hn vs. 1000/T as shown on Fig. 9(d). The values of activation energy of relaxation are listed in Table 3. These values of activation energy for relaxation are similar to that of migration energy which indicates same mechanism for migration as well as for relaxation. The variation of relaxation time τ_​hn with doping concentration of Dy³⁺ ions is shown in the inset of Fig. 9(c) which shows a decrease in relaxation time with doping concentration reaching a minimum value for x = 0.20 and increases again with x. This indicates that, the ion movement becomes faster with doping concentration up to x = 0.20. Since with the doping concentration the density of free ions increases this results the increase in collisions of this free ions with ions of lattice, so relaxation time decreases with x upto x = 0.20. At higher concentration, the ion movement becomes slower which may be due to the strong interaction between ions and oxygen vacancies. Again it should be noticeable that the value of relaxation time for pure ceria is higher than the Dy doped ceria. The master modulus curve or normalized plot at different temperatures for the sample x = 0.20 is shown in Fig. 10(a). Here, all the modulus spectra at different temperatures superimpose on a single master curve pointing to the fact that, it obeys the time–temperature superposition principle (TTSP). In case of imaginary part of the complex modulus spectra, the TTSP can be represented by the following scaling law:F is the scaling function which is independent of temperature. Similar behavior has also been found for other compositions. This scaling behavior indicates the temperature independence of the distribution of relaxation time i.e. relaxation mechanism is temperature independent.³⁷ Fig. 10(b) shows the scaled modulus spectra for different compositions at a particular temperature that indicates the superposition of all the spectra on a single master curve. Therefore, the increase in temperature and doping concentration mainly increase the oxygen vacancy concentration but the relaxation mechanism as mentioned above remains unchanged. We have also calculated the FWHM of the master modulus curve and have found them greater than 1.14 decades. This indicates the non-Debye type nature³⁸ of the compositions, which is well supported by complex impedance plot.

Fig. 9 (a) The complex modulus plots for all the compositions at temperature 525 °C. The variation of (b) real and (c) imaginary part of complex modulus for all the compositions at temperature 525 °C. The variation of relaxation time τ_hn with doping concentration is shown in the inset of (c) and (d) Arrhenius plot of τ_hn for all the compositions.

Table 3 The values of inverse of the high frequency dielectric constant (M_∞), relaxation time τ_hn, shape parameters (μ and ν) at temperature 525 °C and activation energy for relaxation (E_τ) for all compositions. The errors are indicated in the parenthesis

Composition	M∞	τhn (s)	μ (± 0.002)	ν (± 0.01)	Eτ (eV) (± 0.01)
CeO2	0.0046(6)	5.72 × 10^−6	0.048	0.88	1.16
Ce0.9Dy0.1O2−δ	0.0028(8)	1.10 × 10^−6	0.024	0.67	1.05
Ce0.85Dy0.15O2−δ	0.0022(8)	3.52 × 10^−7	0.040	0.89	1.03
Ce0.8Dy0.2O2−δ	0.0022(7)	1.19 × 10^−7	0.048	0.94	1.03
Ce0.75Dy0.25O2−δ	0.0021(4)	1.46 × 10^−7	0.045	0.94	1.04
Ce0.7Dy0.3O2−δ	0.0030(4)	1.61 × 10^−7	0.059	0.93	1.07
Ce0.6Dy0.4O2−δ	0.0041(0)	4.50 × 10^−7	0.058	0.92	1.08
Ce0.5Dy0.5O2−δ	0.0025(9)	5.07 × 10^−6	0.061	0.93	1.11

---

Fig. 10 (a) Scaling of the imaginary part of the complex modulus spectra for Ce_0.8Dy_0.2O_2−δ at different temperatures and (b) modulus master curve for different compositions at a temperature 525 °C.

3.6. AC conductivity

Fig. 11(a) shows the variation of ac conductivity as a function of frequency for all the compositions at temperature 500 °C. The ac conductivity is nearly constant at low frequencies i.e. shows the existence of a plateau (σ_dc) independent on the frequency. At higher frequency ac conductivity increases with frequency and above a certain frequency it increases according to Jonscher's power low³⁹ equation given by

σ′(ω) = σ_dc + Aωⁿ (15)

where A is pre-exponential factor and n is the frequency exponent. The observed conductivity behavior of the compositions can be explained with the jump relaxation model.⁴⁰ Fig. 11(a) also reveals that, the composition Ce_0.8Dy_0.2O_2−δ shows higher dc conductivity. The variation of ac conductivity σ′(ω) and dielectric constant ε′(ω) with doping concentration is shown in Fig. 11(b). As the ac conductivity and dielectric constant varies similarly with doping concentration, therefore they should be closely related to each other. It has been found out that the samples show a variation of conductivity with composition having a maximum at a particular composition. This may be due to the columbic interaction between dopant concentration and oxygen vacancy expressed as,where r is the inter-ionic distance and ε is the dielectric constant of the composition. The above equation indicates that lower value of ε gives higher columbic interaction between oxygen vacancy and dopant ions and consequently higher value of activation or migration energy and lower conductivity. As shown in Fig. 11(b), the composition Ce_0.8Dy_0.2O_2−δ shows the highest value of ε′(ω) at 500 °C and at frequency 7.19 kHz. Therefore, in this composition the columbic interaction is minimum and the conduction is maximum among all other compositions. Again for pure ceria σ′(ω) shows lower value than the Dy doped ceria due to the lower value of ε′(ω) for pure ceria.

Fig. 11 (a) The variation of ac conductivity σ′(ω) as a function of frequency for all the compositions at temperature 500 °C and (b) the variation of ac conductivity σ′(ω) and dielectric constant ε′(ω) with doping concentration.

4. Conclusion

In summary, Dy doped ceria nanomaterials were prepared by low temperature citrate auto-ignition method. Rietveld analysis of the XRD data and HR-TEM image of the sintered samples confirms the well crystalline single phase cubic fluorite structure with space group Fm3̄m. The HR-TEM image showed the presence of lattice distortion in the samples. The impedance spectra were analyzed using suitable RQ circuit and found that both the grain and grain boundary contribution were present in the materials. Impedance analysis also confirmed the non-Debye type relaxation in the materials. The variation of conductivity was correlated with oxygen vacancies and defect associates. Different parameters obtained from the HN fitting of dielectric and modulus spectra were found to be temperature dependent. The shift of the relaxation peak of tan δ with doping concentration has been explained in the light of formation of dimers or trimers and evolution of oxygen vacancies. The relaxation peak in modulus spectra was attributed to the charge re-orientation relaxation of defect pairs. Modulus analysis established the possibility of hopping mechanism for electrical transport process in the system. The conductivity values, dielectric and modulus properties of pure ceria were compared to that of Dy doped ceria. The conductivity values and dielectric properties showed x = 0.20 as the optimum doping concentration in our present system.

Acknowledgements

One of the authors (AD) thankfully acknowledges the financial assistance from Department of Science and Technology (Govt. of India) (Grant no. SR/FTP/PS-141-2010). The authors (SA and AD) also acknowledge the instrumental support from DST (Govt. of India) under departmental FIST programme (Grant no. SR/FST/PS-II-001/2011) and University Grants Commission (UGC) for departmental CAS (Grant no. F.530/5/CAS/2011(SAP-I)) scheme.

References

1. B. C. H. Steele, Solid State Ionics, 2000, 129, 95–110.
2. T. Taniguchi, T. Watanabe, N. Sugiyama, A. K. Subramani, H. Wagata, N. Matsushita and M. Yoshimura, J. Phys. Chem. C, 2009, 113, 19789–19793.
3. A. Ismail, J. B. Giorgi and T. K. Woo, J. Phys. Chem. C, 2012, 116, 704–713.
4. M. Yashima and T. Takizawa, J. Phys. Chem. C, 2010, 114, 2385–2392.
5. M. Mogensen, N. M. Sammes and G. Tompsett, Phys. Solid State Ionics, 2000, 129, 63–94.
6. R. Gerhardt and A. S. Nowick, J. Am. Ceram. Soc., 1986, 69, 641–646.
7. A. K. Baral and V. Sankaranarayanan, Nonoscale Res. Lett., 2010, 5, 637–643.
8. M. O. Zacate, L. Minervini, D. J. Bradfield, R. W. Grimes and K. E. Sickafus, Solid State Ionics, 2000, 128, 243–254.
9. G. B. Balazs and R. S. Glass, Solid State Ionics, 1995, 76, 155–162.
10. H. Inaba and H. Tagawa, Solid State Ionics, 1996, 83, 1–16.
11. T. Mori, R. Buchanan, D. R. Ou, F. Ye, T. Kobayashi, J. D. Kim, J. Zou and J. Drennan, J. Solid State Electrochem., 2008, 12, 841–849.
12. P. Sarkar and P. S. Nicholson, Solid State Ionics, 1986, 21, 49–53.
13. P. Sarkar and P. S. Nicholson, J. Phys. Chem. Solids, 1989, 50, 197–206.
14. T. Paul and A. Ghosh, Mater. Res. Bull., 2014, 59, 416–420.
15. T. Badapandaa, V. Senthila, S. K. Routb, S. Panigrahia and T. P. Sinhac, Mater. Chem. Phys., 2012, 133, 863–870.
16. G. Cao, L. Feng and C. Wang, J. Phys. D: Appl. Phys., 2007, 40, 2899–2905.
17. A. K. Baral and V. Sankaranarayanan, Appl. Phys. A, 2010, 98, 367–373.
18. K. P. Padmasree, R. A. Mantalvo-Lozano, S. M. Montemayor and A. F. Fuentes, J. Alloys Compd., 2011, 509, 8584–8589.
19. H. Yamamura, S. Takeda and K. Kakinuma, Solid State Ionics, 2007, 178, 1059–1064.
20. S. A. Acharya, J. Power Sources, 2012, 198, 105–111.
21. S. Anirban and A. Dutta, J. Phys. Chem. Solids, 2015, 76, 178–183.
22. S. Anirban, T. Paul and A. Dutta, RSC Adv., 2015, 5, 50186–50195.
23. S. Anirban, T. Paul, P. T. Das, T. K. Nath and A. Dutta, Solid State Ionics, 2015, 270, 73–83.
24. D. Pérez-Coll, P. Núñez, J. C. Ruiz-Morales, J. Peña-Martíez and J. R. Frade, Electrochim. Acta, 2007, 52, 2001–2008.
25. S. Ramesh and K. C. James Raju, Int. J. Hydrogen Energy, 2012, 37, 10311–10317.
26. S. Omar, E. D. Wachsman and J. C. Nino, Solid State Ionics, 2008, 178, 1890–1897.
27. S. Havriliak and S. Negami, Polymer, 1967, 8, 161–210.
28. A. K. Baral, Electrochem. Acta, 2010, 56, 667–675.
29. T. Paul and A. Ghosh, J. Appl. Phys., 2014, 116, 144102.
30. M. Biswas, J. Alloys Compd., 2010, 491, 30–35.
31. A. K. Baral, H. P. Dasari, B. K. Kim and J. H. Lee, J. Alloys Compd., 2013, 575, 455–460.
32. P. P. Dholabhai, J. B. Adams, P. Crozier and R. Sharmaw, Phys. Chem. Chem. Phys., 2010, 12, 7904–7910.
33. H. Yamamura, S. Takeda and K. Kakinuma, Solid State Ionics, 2007, 178, 889–893.
34. M. Ram, Phys. B, 2010, 405, 602–605.
35. P. Ganguly, A. K. Jha and K. L. Deori, Solid State Commun., 2008, 146, 472–477.
36. P. Sarkar and P. S. Nicholson, J. Am. Ceram. Soc., 1989, 72, 1447–1449.
37. K. Kathyat, A. Panigrahi, A. Pandey and S. Kar, Mater. Sci. Appl., 2012, 3, 390–397.
38. S. Brahma, R. N. P. Choudhary and A. K. Thakur, Phys. B, 2005, 355, 188–201.
39. A. K. Jonscher, Nature, 1977, 267, 673–679.
40. K. Funke, Solid State Ionics, 1997, 94, 27–33.

---